Convergence in distribution of products of random matrices

Abstract

We consider a sequence A 2, A 2, ... of i.i.d. nonnegative matrices of size d × d, and investigate convergence in distribution of the product M n: =A 1 ... A n. When d≧2 it is possible for M n to converge in distribution (without normalization) to a distribution not concentrated on the zero matrix. Several equivalent conditions for this to happen are given. These lead to a fairly general family of examples. These conditions can also be used to determine when the a.s. limit of 1/nlog∥M n ∥ equals the logarithm of the largest eigenvalue of E(A 1).